Why do farther notes resonate more than closer notes?

Source: 30:05 juncture. Prof. Steven Cassedy. The following is based on YouTube's transcript that lacks formatting, and whose errors I corrected. Sorry for the strange numbering; I originally deleted the time junctures, but then decided to keep them after a few sentences.

How do we prove that there are such things that relies on the principle of sympathetic vibration? If an object is vibrating at a certain frequency, a nearby object object tuned to that same frequency will begin to vibrate together sympathetically with the first object. So let's say I want to prove that this low C on the piano (this one down here) has the next two overtones: that's the C above it and the G above that as part of its overtone series. What I do is: I hold
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down the next C up without playing it
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that releases the damper so that the
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string can vibrate, and then I play the
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lower C as hard as I can and release it
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to release its damper.
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If you're still hearing that upper C,
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it's on the overtone series; it means
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that the lower C had that and made
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it sympathetically vibrate similarly
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with the G above it. And on a great big
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piano like this you can kind of hear it
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resonate. In fact, it resonates for some
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reason, that I don't know, more than the
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C which is closer to it and similarly
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for frequencies farther out on the
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series, but it would be very very
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difficult to hear that on an instrument
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like this even in a hall with the
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acoustics like the ones in Prebys Hall.
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All right. So we don't hear these overtones
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separately when we hear a C played on
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the piano, but they're certainly there.

I ask about the bolded. Please answer simply; I am ignorant in physics or acoustics.

• I don't think the bolded text is correct. It could be that it sounds louder to the professor, or that the particular piano has resonances in its sound board or cabinet that emphasize the G over the C. In general, though, it's not true that higher harmonic numbers resonate better than lower ones. There are specific situations where that might happen, but not generally. – Todd Wilcox Aug 22 '17 at 20:40
• I'd echo Todd. I am quite familiar with the phenomenon bit not the bit about the more distant notes resonating more. One possible situation is that the piano is not quite in tune. Maybe the G was closer to third harmonic than the C is to the second. I am away from home so I cannot test it for a few days. I sometimes demonstrate the phenomenon by getting guitar strings to resonate with notes from the piano. – badjohn Aug 22 '17 at 20:51

1 Answer

Prof Cassidy might not "know the reason," but it should be fairly obvious to any engineer working on vibration measurement and testing that this is what you would expect. (Cassidy is a prof of Slavonic literature, not of engineering!)

From the position of the hammer hitting the string, it's not surprising that for the first few harmonics, more energy goes into the higher harmonics than the lower ones, and so the other string resonates with greater amplitude. This is a fundamental difference between the piano and most other musical instruments.

See the spectrum for note G1 in Fig.5.14 (page 32) of http://www.jjburred.com/research/pdf/burred_acoustics_piano.pdf. For the first 14 harmonics the general trend is increasing amplitude, and the third harmonic has a much bigger amplitude than that general trend.

In fact, the reason why you perceive (with your brain, not just your ears!) a bunch of high harmonics with very little energy at the fundamental frequency as "a low note" and not "a chord containing a lot of high notes" is another hard problem to explain, in the field of psychoacoustics.

Note, for the lowest notes of the piano there are also "inharmonic vibrations" such as axial vibration along the length of the string. These occur because the lowest strings have a complicated structure (two or three layers of coiled material around a thin central core) that doesn't behave the same way as the higher notes, which are produced by simple stretched wires.

Some of these inharmonic vibrations are clearly audible (at relatively high frequencies, and "out of tune" with the note itself) if you play a single low note as loudly as possible. They are one of the reasons why trying to simulate a realistic piano sound "from basic physical principles" is a difficult challenge.

If one of these inharmonic vibrations has a frequency close to the 3rd harmonic of the "real" note, that would explain why the amplitude of the 3rd harmonic appears to be even higher than the general upward trend.

• The spectral charts in the linked document are based on sound pressure levels, which means they are partly the result of resonance, and don't measure the amount of resonant coupling between strings. – Todd Wilcox Aug 22 '17 at 22:19
• @ToddWilcox If you only play a single note, there is no resonant coupling - you are just measuring the excitation. The SPL measures the amount of energy being radiated by each harmonic, which is a reasonable measure of how much energy is available to radiate or couple to something else. Of course you could use a scanning laser Doppler vibrometer to measure the vibrating shape along the complete length of both strings, plus the motion of the soundboard, but that probably isn't cost effective for understanding what's going on - the SLDV would cost about the same as a Steinway concert Grand! – user19146 Aug 22 '17 at 23:40